| L(s) = 1 | − 2·5-s − 2·11-s + 2·13-s − 6·17-s − 19-s − 2·23-s − 25-s − 4·29-s − 8·31-s − 2·37-s + 8·41-s − 8·43-s − 2·47-s − 7·49-s + 4·53-s + 4·55-s + 2·61-s − 4·65-s + 12·67-s + 4·71-s + 6·73-s − 16·79-s − 6·83-s + 12·85-s + 2·95-s − 2·97-s + 10·101-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.603·11-s + 0.554·13-s − 1.45·17-s − 0.229·19-s − 0.417·23-s − 1/5·25-s − 0.742·29-s − 1.43·31-s − 0.328·37-s + 1.24·41-s − 1.21·43-s − 0.291·47-s − 49-s + 0.549·53-s + 0.539·55-s + 0.256·61-s − 0.496·65-s + 1.46·67-s + 0.474·71-s + 0.702·73-s − 1.80·79-s − 0.658·83-s + 1.30·85-s + 0.205·95-s − 0.203·97-s + 0.995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 3 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.12195161937843202649005513173, −9.057006068524209439978003598025, −8.293866406473810377354005199200, −7.49672369325431092785641599212, −6.58468415485848449567440294355, −5.48044237465026571070978672561, −4.34892306129675184831014993846, −3.51808670484264273930118040379, −2.07210083625649444067550413290, 0,
2.07210083625649444067550413290, 3.51808670484264273930118040379, 4.34892306129675184831014993846, 5.48044237465026571070978672561, 6.58468415485848449567440294355, 7.49672369325431092785641599212, 8.293866406473810377354005199200, 9.057006068524209439978003598025, 10.12195161937843202649005513173
